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Drinfeld Centers and Morita Equivalence Classes of Fusion 2-Categories

Thibault D. Décoppet

TL;DR

This work establishes that Drinfeld centers of fusion 2-categories are Morita invariant and finite semisimple, connecting higher categorical centers to Witt theory. It proves that Morita equivalence classes of connected fusion 2-categories correspond to symmetric centers and Witt data, and shows every fusion 2-category is Morita equivalent to a 2-Deligne tensor product of a strongly fusion 2-category with an invertible one, further reducing to connected cases. The paper also proves that every rigid algebra in a fusion 2-category is separable, implying separability for all multifusion 2-categories, and defines a nonzero dimension for fusion 2-categories, aligning with Koszul-like finiteness conditions. Collectively, these results yield a robust 4-categorical Morita theory for fusion 2-categories and yield strong structural consequences for their centers, separability properties, and potential 4-dualizability in extended TQFT contexts.

Abstract

We prove that the Drinfeld center of a fusion 2-category is invariant under Morita equivalence. We go on to show that the concept of Morita equivalence between connected fusion 2-categories recovers exactly the notion of Witt equivalence between braided fusion 1-categories. A strongly fusion 2-category is a fusion 2-category whose braided fusion 1-category of endomorphisms of the monoidal unit is $\mathbf{Vect}$ or $\mathbf{SVect}$. We prove that every fusion 2-category is Morita equivalent to the 2-Deligne tensor product of a strongly fusion 2-category and an invertible fusion 2-category. We proceed to show that every fusion 2-category is Morita equivalent to a connected fusion 2-category. As a consequence, we find that every rigid algebra in a fusion 2-category is separable. This implies in particular that every fusion 2-category is separable. Conjecturally, separability ensures that a fusion 2-category is 4-dualizable. We define the dimension of a fusion 2-category, and prove that it is always non-zero. Finally, we show that the Drinfeld center of any fusion 2-category is a finite semisimple 2-category.

Drinfeld Centers and Morita Equivalence Classes of Fusion 2-Categories

TL;DR

This work establishes that Drinfeld centers of fusion 2-categories are Morita invariant and finite semisimple, connecting higher categorical centers to Witt theory. It proves that Morita equivalence classes of connected fusion 2-categories correspond to symmetric centers and Witt data, and shows every fusion 2-category is Morita equivalent to a 2-Deligne tensor product of a strongly fusion 2-category with an invertible one, further reducing to connected cases. The paper also proves that every rigid algebra in a fusion 2-category is separable, implying separability for all multifusion 2-categories, and defines a nonzero dimension for fusion 2-categories, aligning with Koszul-like finiteness conditions. Collectively, these results yield a robust 4-categorical Morita theory for fusion 2-categories and yield strong structural consequences for their centers, separability properties, and potential 4-dualizability in extended TQFT contexts.

Abstract

We prove that the Drinfeld center of a fusion 2-category is invariant under Morita equivalence. We go on to show that the concept of Morita equivalence between connected fusion 2-categories recovers exactly the notion of Witt equivalence between braided fusion 1-categories. A strongly fusion 2-category is a fusion 2-category whose braided fusion 1-category of endomorphisms of the monoidal unit is or . We prove that every fusion 2-category is Morita equivalent to the 2-Deligne tensor product of a strongly fusion 2-category and an invertible fusion 2-category. We proceed to show that every fusion 2-category is Morita equivalent to a connected fusion 2-category. As a consequence, we find that every rigid algebra in a fusion 2-category is separable. This implies in particular that every fusion 2-category is separable. Conjecturally, separability ensures that a fusion 2-category is 4-dualizable. We define the dimension of a fusion 2-category, and prove that it is always non-zero. Finally, we show that the Drinfeld center of any fusion 2-category is a finite semisimple 2-category.
Paper Structure (19 sections, 42 theorems, 43 equations)

This paper contains 19 sections, 42 theorems, 43 equations.

Table of Contents

  1. Preliminaries
  2. Finite Semisimple 2-Categories
  3. Fusion 2-Categories
  4. Rigid and Separable Algebras
  5. The Morita Theory of Fusion 2-Categories
  6. The Drinfeld Center
  7. Definition
  8. Elementary Properties
  9. Morita Invariance
  10. The Morita Theory of Connected Fusion 2-Categories
  11. Comparison with Witt equivalence
  12. Technical Results
  13. Fusion 2-Categories up to Morita Equivalence
  14. Strongly Fusion and Invertible
  15. Connected
  16. ...and 4 more sections

Key Result

Theorem 1.3.3

Let $A$ be a rigid algebra in a fusion 2-category $\mathfrak{C}$. Then, $A$ is separable if and only if $\mathbf{Bimod}_{\mathfrak{C}}(A)$, the 2-category of $A$-$A$-bimodules in $\mathfrak{C}$, is finite semisimple. Further, if either of these conditions is satisfied, $\mathbf{Mod}_{\mathfrak{C}}(A

Theorems & Definitions (118)

  • Definition 1.1.1
  • Definition 1.1.2
  • Definition 1.1.3
  • Definition 1.1.4
  • Definition 1.1.5
  • Definition 1.2.1
  • Example 1.2.2
  • Example 1.2.3
  • Example 1.2.4
  • Definition 1.2.5
  • ...and 108 more